A Theory of Shape Reconstruction from Heat Conduction and Shading

2026-10-02

AI Summary

This study addresses the conical ambiguity inherent in single-image shadow-based shape reconstruction and the local convex-concave ambiguity encountered in heat conduction-based shape estimation. To overcome these limitations, this work proposes a novel theoretical framework that integrates shadow cues with the heat conduction equation. Through Laplacian estimation and heat transport modeling, the method jointly resolves multimodal geometric constraints under unknown illumination conditions. Crucially, it disambiguates shape solely through physical consistency, eliminating the need for conventional smoothness priors. The effectiveness of the proposed approach is validated on both complex synthetic shapes and noisy real-world thermal videos. Demonstrating robust performance against albedo variations, this research establishes a new paradigm for cross-modal three-dimensional reconstruction.

Abstract

Shape from shading using a single image of a Lam- bertian surface is inherently ambiguous. When the light source direction is known, the surface normal estimation has a cone- ambiguity, which worsens when the source is unknown. Recently, shape from heat conduction has emerged as an approach that leverages heat transport equations to estimate the Shape Lapla- cian operator, an intrinsic measure of shape. However, deriving surface normals from the Laplacian operator encounters a local binary convex/concave ambiguity. Our contribution introduces a novel theory to resolve these local shape ambiguities (excluding a few degeneracies) without relying on priors like smoothness, by combining the cues from shading and heat conduction. Our method ensures the mathematical constraints of both shading and the Laplacian are satisfied simultaneously, even with an unknown light source. We validate our theory through simulations of complex shapes and analyze its performance in the presence of noise, as well as on a noisy single thermal video of real-world objects with complex shapes and material properties, including varying albedo.

Problem

• shape from shading
• shape from heat conduction
• surface normal ambiguity
• convex/concave ambiguity
• Laplacian operator

Innovation

• Shape from shading
• Heat conduction
• Shape Laplacian
• Ambiguity resolution
• Surface normal estimation